Noah K. Murray/The Star-LedgerGiants first-round pick and former Nebraska CB Prince Amukamara is surrounded by his family, royalty in Nigeria, as he poses for cameras tonight at Radio City Music Hall.

The Giants say over and over and over they stick to their draft board and select players according to value, not need.

Maybe it’s time to start believing them.

Tonight, with the 19th overall pick, they drafted Nebraska cornerback Prince Amukamara, a player they fully expected to be long gone by the time their selection rolled around. And yes, we know that’s one of the go-to clichés on draft night — that a team had no idea the guy they selected would be there — but in this case that’s the truth because plenty of mock drafts had Amukamara going in the top 10.

Though the Giants could have used an offensive lineman or a running back, they went with the player that was “clearly the highest-rated player on the board,” according to coach Tom Coughlin.

“It happens like that sometimes in the draft — guys can fall right into your lap, and we think that’s what happened,” general manager Jerry Reese said. “Everybody’s excited up there. Our defensive staff’s excited up there; our entire room is excited about him.”

The 6-foot, 206-pound Amukamara was a unanimous All-America selection this past season, despite the fact he recorded zero in terceptions. In fact, even Reese admitted that led to “some concern about his production on the ball” because he had fallen off from grabbing five interceptions as a junior — in part thanks to the presence of defensive tackle Ndamukong Suh, the second overall pick of the Lions last year.

But Amukamara was very much a factor in the Cornhuskers’ secondary last season, deflecting 13 passes and limiting opposing offenses to a 35 percent completion rate in passes thrown his way.

Said Reese, “He’s (got) height, weight, speed, this guy’s big and he’s fast. We think he can still play the ball really well. He’s a good tackler, a physical player. So there are a lot of things to like about Prince.”

Unless you’re Aaron Ross. The selection of Amukamara means Giants’ first-round pick in 2007 might very well be entering his final year with the team. Starting cornerback Corey Webster is signed through 2013 and fellow corner Terrell Thomas joins Ross as a free agent after this season. The presence of all three players means it could be tough for Amukamara to get a lot of playing time this upcoming season.

But like the Giants showed when they drafted defensive ends Mathias Kiwanuka (2006) and Jason Pierre-Paul (2010), it’s not always about next year for them. It’s about the next several years.

“He was too much value, and it’s a need pick as well for me,” Reese said of Amukamara. “He’ll play right away; he’ll be in the mix in our secondary right away.”

Added Coughlin, “You can never have enough cover guys.”

Sure, the Giants needed a running back and could have selected Alabama running back and 2009 Heisman winner Mark Ingram, whom they liked very much. And they even could have grabbed Boston College offensive tackle Anthony Castonzo, though their grades on him weren’t as high as many had projected.

Colorado offensive tackle Nate Solder went to the Patriots at No. 17. Had he still been on the board, it would have been interesting because the Giants were enamored with him.

But not as much as they liked Amukamara, even if he fell somewhat inexplicably.

“There is always a concern, but we go by what we have on him,” Reese said of Amukamara’s slide, adding: “We thought he was a terrific player and we never dreamed he would be there at that point.”